$J$-Domination in Graphs

Abstract

Let $G$ be a graph. A subset $D=\{d_1, d_2, \cdots, d_m\}$ of vertices of $G$ is called a $J$-set if $N_G[d_i]\setminus N_G[d_j]\neq \varnothing$ for every $i\neq j$, where $i,j\in\{1, 2, \ldots, m\}$. A $J$-set is called a $J$-dominating set of $G$ if $D=\{d_1, d_2, \ldots, d_m\}$ is a dominating set of $G$. The $J$-domination number of $G$, denoted by $\gamma_{J}(G)$, is the maximum cardinality of a $J$-dominating set of $G$. In this paper, we introduce this new concept and we establish formulas and properties on some classes of graphs and in join of two graphs. Upper and lower bounds of $J$-domination parameter with respect to the order of a graph and other parameters in graph theory are obtained. In addition, we present realization result involving this parameter and the standard domination. Moreover, we characterize $J$-dominating sets in some classes of graphs and join of two graphs and finally determine the exact value of the parameter of each of these graphs.